# Homoclinism of Lie rings

From Groupprops

## Definition

### Short definition

A **homoclinism of Lie rings** is a homologism of Lie rings with respect to the subvariety of the variety of Lie rings given by abelian Lie rings.

### Full definition

For any Lie ring , let denote the center of , denote the Lie ring of inner derivations of (explicitly, it is isomorphic to , and denote the derived subring of .

Let denote the mapping that arises from the Lie bracket mapping , and then observing that this map is constant on the cosets of . Note that the mapping is -bilinear.

A **homoclinism of Lie rings** and is a pair where is a homomorphism of with <math\operatorname{Inn}(M)</math> and is a homomorphism of with , such that .

## Related notions

Term | Meaning |
---|---|

category of Lie rings with homoclinisms | category whose objects are Lie rings and whose morphisms are homoclinisms |

isoclinism of Lie rings | an invertible homoclinism, or equivalently, a homoclinism where both the component homomorphisms are isomorphisms |

homoclinism of groups | the analogous concept for groups |

homologism of Lie rings | a more general concept of which homoclinisms are a special case. Homoclinisms are homologisms with respect to the subvarety of abelian groups in the variety of Lie rings. |